Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{2n^2 - 22n + 36}{-5n^2 + 40n - 60}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {2(n^2 - 11n + 18)} {-5(n^2 - 8n + 12)} $ $ p = -\dfrac{2}{5} \cdot \dfrac{n^2 - 11n + 18}{n^2 - 8n + 12} $ Next factor the numerator and denominator. $ p = - \dfrac{2}{5} \cdot \dfrac{(n - 2)(n - 9)}{(n - 2)(n - 6)}$ Assuming $n \neq 2$ , we can cancel the $n - 2$ $ p = - \dfrac{2}{5} \cdot \dfrac{n - 9}{n - 6}$ Therefore: $ p = \dfrac{ -2(n - 9)}{ 5(n - 6)}$, $n \neq 2$